How to use for loops to calculate the determinant of the first n powers of 2x2 matrix (A) without using the implicit Matlab command "det"

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Blake 2015-9-28

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评论： Blake 2015-9-29

I am allowed to use the for loop as well as if/elseif/else statements to create the function but I am not sure how exactly to do this. The input will be a matrix A and a scalar value n. I began using if statements to make sure that the matrix is 2x2 and that n is positive however i do not know how to code for det(A^n) without using the det function. Below is an example of what i have thus far:

 function ret = invertiblePowers(A,n)
   if isequal(size(A), [2 2])==0
      ret= disp('Matrix has wrong dimensions')
   elseif floor(n)~=ceil(n)
      ret= disp('n is not a positive integer')
   elseif  isequal(size(A), [2 2])==1 & floor(n)=ceil(n)

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Answer 1

WAT 2015-9-28

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在 MATLAB Online 中打开

You're going to want a function that takes A and n as inputs and either returns a string or nothing at all. That would look like

function ret = invertiblePowers(A,n)

or

function [] = invertiblePowers(A,n)

Assuming you want to return a string, then just do something like

 function ret = invertiblePowers(A,n)
 ret = ''; % initialize ret to empty string
 % Make sure A is the right size
 if (~isequal(size(A),[2,2]))
  ret = 'Matrix has wrong dimensions'; % note that 'disp' is not used
  return;  % stop working and exit out of this function
 end
 % Make sure n is positive integer
 if ((floor(n)~=ceil(n)) || (n <=0))
   ret = 'n is not a positive integer';
   return;
 end
 % if A is 2x2 and n is positive integer, find det(A^n)
 % It's unclear whether this function needs to find all 
 %  determinates for I=1 up to n, or just n. 
 % Assuming you want 1:n, loop through
 for i = 1:n
   An = A^i; % raise A to the ith power
   % find the determinate here, you need to do this part
   detAn = YOUR MATH HERE;
   % append results to return string
   ret = [ret ; sprintf('n = %i : det(A^n) = %f',i,detAn)];
 end % end for loop
 end

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Blake 2015-9-29

This makes sense! Thank you!

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Answer 2

Walter Roberson 2015-9-28

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http://math.stackexchange.com/questions/693948/simple-way-to-find-the-sign-of-a-determinant-given-a-singular-value-decompositio

This together with the fact that with SVD, the N'th power of the matrix can be found by taking the N'th power of the diagonal.

Or you could just use the formula for the determinant of a 2 x 2 matrix.

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Blake 2015-9-28

在 MATLAB Online 中打开

Ok so i tried switching to using the formula for a det(A) and I got rid of the extra elseif statments since those would be filtered out already. I still am unsure how to incorporate displaying the strings. The next part of this asks me to call the function and evaluate a provided matrix and n value.

 function [] = invertiblePowers(A,n)
%UNTITLED6 Summary of this function goes here
%   Detailed explanation goes here
if isequal(size(A), [2 2])==1 & n>0
    (A(1)*A(4)-A(2)*A(3))^n
elseif (A(1)*A(4)-A(2)*A(3))^n==0
    disp('det(A^k) = 0 so A^k is singular')
elseif (A(1)*A(4)-A(2)*A(3))^n~=0
    disp('det(A^k) = A_k so A^k is invertible')

end

WAT 2015-9-28

I wouldn't assume you're allowed to use the rule that det(A^n) = det(A)^n. I'd assume you're supposed to calculate A^n then find the determinate of that new matrix.

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Answer 3

James Tursa 2015-9-28

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To calculate the determinant of a 2x2 matrix, see this link a little over halfway down the first page:

http://mathworld.wolfram.com/Determinant.html

To get the determinant of a matrix power, det(A^n), also note from the above link that the determinant of a matrix product is the product of the individual determinants. I.e. det(A*A) = det(A)*det(A). So you can extend this to powers and figure out the formula for det(A^n).

Using the above hints should help you to write the code.

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WAT 2015-9-28

I'm guessing that discovering that relationship between det(A^n) and det(A) is the point of this problem =P

Blake 2015-9-28

I see the relationship that det(A^n) can be determined by finding the det(A) then raising it to the nth power but I am confused about incorporating the for loop. In the comment section of the other response you can see I'm only using if and elseif statements

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